The Schur–Szegö composition for hyperbolic polynomials

Abstract

The composition of Schur–Szegö of the polynomials P ( x ) = ∑ j = 0 n C n j a j x j and Q ( x ) = ∑ j = 0 n C n j b j x j is defined as P ∗ Q = ∑ j = 0 n C n j a j b j x j . In the case when P and Q are hyperbolic, i.e. with real roots only, we give the exhaustive answer to the question if the numbers of positive, negative and zero roots of P and Q are known what these numbers can be for P ∗ Q . To cite this article: V.P. Kostov, C. R. Acad. Sci. Paris, Ser. I 345 (2007).